I am new to the definition of Hausdorff, but when I look at it, I would try to find open sets which include the components of these subsets, see if their intersections equal the empty set.
However, I am not sure which open sets I can use?
For example, is it too simplistic to say that 1 is not Hausdorff, because open set {a} intersecting with open set {a,c,d} = {a} which does not equal the empty set?

I am new to the definition of Hausdorff, but when I look at it, I would try to find open sets which include the components of these subsets, see if their intersections equal the empty set.
However, I am not sure which open sets I can use?
For example, is it too simplistic to say that 1 is not Hausdorff, because open set {a} intersecting with open set {a,c,d} = {a} which does not equal the empty set?

No, that is not quite the idea, unless I miss read you.
Given any two points, you must be able to find two disjoint open sets each containing one of the points. Because the topologies are finite, it is easy to check this out.

Collection 1 fails under the union condition as the two singleton subsets implies {a,b} must also be present?
Collection 3 because the intersection condition fails, {a,b} and {b,c,d} would give the singleton subset {b} ?

Collection 1 fails under the union condition as the two singleton subsets implies {a,b} must also be present? BUT it is.
Collection 3 because the intersection condition fails, {a,b} and {b,c,d} would give the singleton subset {b} ?You don't see that {b} is there?

No, that is not quite the idea, unless I miss read you.
Given any two points, you must be able to find two disjoint open sets each containing one of the points. Because the topologies are finite, it is easy to check this out.

Therefore, since all three collections are topologies, the subsets in each case 1,2 & 3 are called the open subsets of X.
But none of these can be Hausdorff since any distinct pair of points from X, e.g. a & b contained in any of the sets in 1,2 or 3, intersected with X will not be empty...is that a correct viewpoint?